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We give a geometric invariant theory (GIT) construction of the log canonicalmodel $\bar M_g(\alpha)$ of the pairs $(\bar M_g, \alpha \delta)$ for $\alpha\in (7/10 - \epsilon, 7/10]$ for small $\epsilon \in \mathbb Q_+$. We show that$\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow varietybicanonical curves; $\bar M_g(7/10-\epsilon)$ is isomorphic to the GIT quotientof the asymptotically-linearized Hilbert scheme of bicanonical curves. In eachcase, we completely classify the (semi)stable curves and their orbit closures.Chow semistable curves have ordinary cusps and tacnodes as singularities but donot admit elliptic tails. Hilbert semistable curves satisfy further conditions,e.g., they do not contain elliptic bridges. We show that there is a smallcontraction $\Psi: \bar M_g(7/10+\epsilon) \to \bar M_g(7/10)$ that contractsthe locus of elliptic bridges. Moreover, by using the GIT interpretation of thelog canonical models, we construct a small contraction $\Psi^+ : \barM_g(7/10-\epsilon) \to \bar M_g(7/10)$ that is the Mori flip of $\Psi$.

Log minimal model program for the moduli space of stable curves: The first flip